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Statistics & Decisions 99, 9019999 (2004)c R. Oldenbourg Verlag, M unchen 2004

An Asymptotic Analysis of the Mean-VariancePortfolio Selection

Gy orgy Ottucs ak, Istv an Vajda

Received: Month-1 99, 2003; Accepted: Month-2 99, 2004

Summary: This paper gives an asymptotic analysis of the mean-variance (Markowitz-type) port-folio selection under mild assumptions on the market behavior. Theoretical results show the rate of underperformance of the risk aware Markowitz-type portfolio strategy in growth rate compared tothe log-optimal portfolio strategy, which does not have explicit risk control. Statements are givenwith and without full knowledge of the statistical properties of the underlying process generatingthe market, under the only assumption that the market is stationary and ergodic. The experimentsshow how the achieved wealth depends on the coefcient of absolute risk aversion measured onpast NYSE data.

1 IntroductionThe goal of the Markowitzs portfolio strategy is the optimization of the asset allocationin nancial markets in order to achieve optimal trade-off between the return and the risk (variance). For the static model (one-period model), the classical solution of the mean-variance optimization problem was given by Markowitz [16] and Merton [17]. Comparedto expected utility models , it offers an intuitive explanation for diversication. However,most of the mean-variance analysis handles only static models contrary to the expectedutility models, whose literature is rich in multiperiod models .

In the multiperiod models (investment strategies) the investor is allowed to rebalancehis portfolio at the beginning of each trading period. More precisely, investment strate-gies use information collected from the past of the market and determine a portfolio, thatis, a way to distribute the current capital among the available assets. The goal of the in-vestor is to maximize his utility in the long run. Under the assumption that the daily pricerelatives form a stationary and ergodic process the asymptotic growth rate of the wealthhas a well-dened maximum which can be achieved in full knowledge of the distribution

of the entire process, see Algoet and Cover [3]. This maximum can be achieved by thelog-optimal strategy , which maximizes the conditional expectation of log-utility .

AMS 1991 subject classication: 90A09, 90A10Key words and phrases: sequential investment, kernel-based estimation, mean-variance investment, log-optimalinvestment


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For an investor plausibly arises the following question: how much the loss in theaverage growth rate compared to the asymptotically best rate if a mean-variance portfoliooptimization is followed in each trading period.

The rst theoretical result connecting the expected utility and mean-variance analysiswas shown by Tobin [19], for quadratic utility function. Grauer [9] compared the log-optimal strategies and the mean-variance analysis in the one-period model for variousspecications of the state return distributions and his experiments revealed that these twoportfolios have almost the same performance when the returns come from the normaldistribution. Kroll, Levy and Markowitz [15] conducted a similar study. Merton [17] de-veloped a continuous time mean-variance analysis and showed that log-optimal portfoliois instantaneously mean-variance efcient when asset prices are log-normal. Hakanssonand Ziemba [13] followed another method in dening a dynamic mean-variance setting,where the log-optimal portfolio can be chosen as a risky mutual fund. They considereda nite time horizon model and the asset price behavior was determined by a Wiener-process.

In this paper, we follow a similar approach to [17], however in discrete time settingfor general stationary and ergodic processes.

To determine a Markowitz-type portfolio, knowledge of the innite dimensional sta-tistical characterization of the process is required. In contrast, for log-optimal portfoliosetting universally consistent strategies are known, which can achieve growth rate achiev-able in the full knowledge of distributions, however without knowing these distributions.These strategies are universal with respect to the class of all stationary and ergodic pro-cesses as it was proved by Algoet [1]. Gy or and Schafer [11], Gyor, Lugosi, Udina[10] constructed a practical kernel based algorithm for the same problem.

We present a strategy achieving the same growth rate as the Markowitz-type strategyfor the general class of stationary and ergodic processes, however without assuming theknowledge of the statistical characterization of the process. This strategy is risk averse

in the sense that in each time period it carries out the mean-variance optimization.We also present an experimental performance analysis for the data sets of New York Stock Exchange ( NYSE ) spanning a twenty-two-year period with thirty six stocks in-cluded, which set was presented in [10].

The rest of the paper is organized as follows. In Section 2 the mathematical modelis described. The investigated portfolio strategies are dened in Section 3. In the nextsection a lower bound on the performance of Markowitz-type strategy is shown. Section5 presents the kernel-based Markowitz-type sequential investment strategy and its mainconsistency properties. Numerical results based on NYSE data set are shown in Section6. The proofs are given in Section 7.

2 Setup, the market modelThe model of stock market investigated in this paper is the one considered, among others,by Breiman [7], Algoet and Cover [3]. Consider a market of d assets. A market vector x = ( x(1) , . . . , x (d ) ) R d+ is a vector of d nonnegative numbers representing pricerelatives for a given trading period. That is, the j -th component x( j ) 0 of x expresses


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An Asymptotic Analysis of the Mean-Variance Portfolio Selection 903

the ratio of the opening prices of asset j . In other words, x( j ) is the factor by whichcapital invested in the j -th asset grows during the trading period.

The investor is allowed to diversify his capital at the beginning of each trading pe-riod according to a portfolio vector b = ( b(1) , . . . b(d ) ). The j -th component b( j ) of bdenotes the proportion of the investors capital invested in asset j . Throughout the paperwe assume that the portfolio vector b has nonnegative components with dj =1 b

( j ) = 1 .The fact that dj =1 b

( j ) = 1 means that the investment strategy is self nancing and con-sumption of capital is excluded. The non-negativity of the components of b means thatshort selling and buying stocks on margin are not permitted. Let S 0 denote the investorsinitial capital. Then at the end of the trading period the investors wealth becomes

S 1 = S 0d

j =1

b( j ) x( j ) = S 0 b , x ,

where , denotes inner product.

The evolution of the market in time is represented by a sequence of market vectorsx 1 , x 2 , . . . R d+ , where the j -th component x

( j )i of x i denotes the amount obtained

after investing a unit capital in the j -th asset on the i-th trading period. For j i weabbreviate by x ij the array of market vectors (x j , . . . , x i ) and denote by d the simplexof all vectors b R d+ with nonnegative components summing up to one. An investment strategy is a sequence B of functions

b i : R d+i 1

d , i = 1 , 2, . . .

so that b i (x i 11 ) denotes the portfolio vector chosen by the investor on the i-th tradingperiod, upon observing the past behavior of the market. We write b (x i 11 ) = b i (x

i 11 )

to ease the notation.Starting with an initial wealth S 0 , after n trading periods, the investment strategy B

achieves the wealth

S n = S 0n

i =1

b (x i 11 ) , x i = S 0eP ni =1 log b (x

i 11 ) , x i = S 0enW n (B ) .

where W n (B ) denotes the average growth rate

W n (B )def =

1n

n

i =1log b (x i 11 ) , x i .

Obviously, maximization of S n = S n (B ) and maximization of W n (B ) are equivalent.In this paper we assume that the market vectors are realizations of a random pro-

cess, and describe a statistical model. Our view is completely nonparametric in that theonly assumption we use is that the market is stationary and ergodic, allowing arbitrar-ily complex distributions. More precisely, assume that x 1 , x 2 , . . . are realizations of therandom vectors X 1 , X 2 , . . . drawn from the vector-valued stationary and ergodic process{X n } . The sequential investment problem, under these conditions, has been consid-ered by, e.g., Breiman [7], Algoet and Cover [3], Algoet [1, 2], Gy or and Sch afer [11],Gyor, Lugosi, Udina [10].


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3 The Markowitz-type and the log-optimal portfolio se-lection

3.1 The Markowitz-type portfolio selectionIn his seminal paper Markowitz [16] used expected utility and the investigation was re-stricted to single period investment with i.i.d returns. In contrary we work with a multi-period model (investment strategy) and we assume general stationary and ergodic modelfor the market returns. Consequently it is more adequate for us to use conditional ex-pected utility. In the special case of i.i.d. returns it is simplied to the standard expectedutility. For the sake of the distinction from the standard Markowitz approach we will callour utility function Markowitz-type utility function .

Let the following formula dene the conditional expected value of the Markowitz-type utility function:

E U M ( b (Xn 11 ) , X n , )|X

n 11

def

= E { b (Xn 11 ) , X n |X

n 11 }

Var b (X n 11 ) , X n |Xn 11 ,

where X n is the market vector for n -th day, b (X n 11 ) d and [0, ) is the con-stant coefcient of absolute risk aversion of the investor. The conditional expected valueof the Markowitz-type utility function can be expressed after some algebra in followingform:

E {U M ( b (X n 11 ) , X n , )|Xn 11 }

= (1 2)E b (X n 11 ) , X n 1|Xn 11 E {( b (X

n 11 ) , X n 1)

2 |X n 11 }

+1 + E 2 b (X n 11 ) , X n |Xn 11 .

Accordingly let the Markowitz-type utility function be dened as follows

U M ( b (X n 11 ) , X n , )def = (1 2)( b (X n 11 ) , X n 1) ( b (X

n 11 ) , X n 1)

2 + 1 + E 2{ b (X n 11 ) , X n |X

n 11 }.

Hence the Markowitz-type portfolio strategy be dened by B = {b ()}, where

b (Xn 11 ) = argmax

b dE U M ( b (X n 11 ) , X n , ) |X

n 11 .

Let S n, = S n (B ) denote the capital achieved by Markowitz-type portfolio strategyB

, after n trading periods.

3.2 The log-optimal portfolio selectionNow we briey introduce the log-optimal portfolio selection. The fundamental limits,rst published in [3], [1, 2] reveal that the so-called log-optimal portfolio B = {b()}


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is the best possible choice for the maximization of S n . More precisely, on trading periodn let b() be such that

b(Xn

11 ) = arg maxb d E log b (Xn

11 ) , X n Xn

11 .

If S n = S n (B ) denotes the capital achieved by a log-optimal portfolio strategy B , aftern trading periods, then for any other investment strategy B with capital S n = S n (B ) andfor any stationary and ergodic process {X n } ,

limsupn

1n

logS nS n

0 a.s.

and

limn

1n

log S n = W a.s.,

where

W = E log b(X 1 ) , X 0is the maximal possible growth rate of any investment strategy. Thus, (almost surely) noinvestment strategy can have a higher growth rate than a log-optimal portfolio.

3.3 Connection of the Markowitz-type and the log-optimal portfolio:an intuitive argument

As the main tool we apply the semi-log function introduced by Gy or, Urb an and Vajda[12]. The semi-log function is the second order Taylor expansion of log z at z = 1

h(z) def = z 1 12

(z 1)2 .

The semi log-optimal portfolio strategy is B = {b}, where

b(X n 11 )def = arg max

b dE h b (X n 11 ) , X n |X

n 11 .

Applying the semi-log approximation we get:

E log b (X n 11 ) , X n Xn 11

E h b (X n 11 ) , X n |Xn 11

= E b (X n 11 ) , X n 1 12

b (X n 11 ) , X n 12

X n 11 . (3.1)

According to formula (3.1) we introduce a few simplifying notations for the conditional

expected valueE n (b )

def = E b (X n 11 ) , X n Xn 11 ,

for the conditional second order moment

E n (b )2def = E b (X n 11 ) , X n

2X n 11 ,


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and nally for the variance

V n (b )def = E n (b )2 E 2n (b ).

Hence we get

b(X n 11 ) = argmaxb d

2E n (b ) 12

E n (b )2 32

= arg maxb d

2E n (b ) 12

E n (b )2 E 2n (b ) 12

E 2n (b )

= arg maxb d

(E n (b )(4 E n (b )) V n (b ))

= arg maxb d

E n (b )n

V n (b )def = b n (X

n 11 ),

where

ndef =

14 E n (b )

is the coefcient of risk aversion of the investor, which is here a function of conditionalexpected value. Note that parameter n dynamically changes in time, which means as if the Markowitzs investor would dynamically adjust his coefcient of absolute risk aver-sion to the past performance of the portfolio.

So far we sketched a basic relationship between the Markowitz-type and semi-logportfolio selection. The relationship between the log-optimal and semi-log optimal ap-proach was examined in [12]. We present formal, rigorous analysis for the comparisonof the investment strategies in the subsequent sections.

4 Comparison of Markowitz-type and log-optimal port-folio selection in case of known distribution

Naturally arises the question, how much we lose in the long run if we are risk averseinvestors compared to the log-optimal investment. The theorem in this section showsan upper bound on the loss in case of known distribution, if our hypothetical investor isrisk averse with parameter . More precisely for an arbitrary we give the growth rateof the Markowitz-type strategy compared to the maximal possible growth rate (which isachieved by the log-optimal investment) in asymptotic sense.

The only assumptions, which we use in our analysis is that the market vectors {X n }come from a stationary and ergodic process, for which

a X ( j )n 1a

(4.1)

for all j = 1 , . . . , d , where 0 < a < 1.


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Theorem 4.1 For any stationary and ergodic process {X n } , for which (4.1) holds, for all 0, 12

W liminf n

1n

log S n,

W +2a 1

1 2E min

mE 1 + log( X (m )0 ) X

(m )0 X

1

2 121 2

E maxm

E (X (m )0 1)2 X 1 minm

E |X (m )0 1| X 1

2

a 3 + 1

3(1 2)E max

mE X (m )0 1

3X 1 a.s.

Remark 4.2 Under the assumption of Theorem 4.1 with slight modication of the proof one can show same result for all 12 , .

Remark 4.3 We have made some experiments on pairs of stocks of NYSE . For IBM andCoca-Cola we got the following estimates on the magnitude of the terms on the right-hand side of the statement of Theorem 4.1 respectively 10 5 , 10 4 , 10 6 . Furthermorethe order of W is 9 10 4 . With optimized one could push the difference betweenW and lim inf n 1n log S

n, below 1% of W .

5 Nonparametric kernel-based Markowitz-type strategyTo determine a Markowitz-type portfolio, knowledge of the innite dimensional statis-tical characterization of the process is required. In contrast, for log-optimal portfoliosetting universally consistent strategies are known, which can achieve growth rate achiev-able in the full knowledge of distributions, however without knowing these distributions.

Roughly speaking, an investment strategy B is called universally consistent with respectto a class of stationary and ergodic processes {X n } , if for each process in the class,

limn

1n

log S n (B ) = W a.s.

The surprising fact that there exists a strategy, universally consistent with respect tothe class of all stationary and ergodic processes with E | log X ( j ) | < for all j =1, . . . , d , was rst proved by Algoet [1] and by Gy or and Schafer [11]. Gyor, Lugosi,Udina [10] introduced kernel-based strategies, here we describe a moving-window ver-sion, corresponding to an uniform kernel function in order to keep the notations simple.

Dene an innite array of experts H (k, ) = {h (k, ) ()}, where k, are positive inte-gers. For xed positive integers k, , choose the radius r k, > 0 such that for any xed

k, lim

r k, = 0 .

Then, for n > k + 1 , dene the expert h (k, ) as follows. Let J n be the locations of matches:

J n = k < i < n : x i 1i k xn 1n k r k, ,


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where denotes the Euclidean norm. Put

h (k, ) (x n 11 ) = arg maxb d { iJ n }

b , x i , (5.1)

if the product is non-void, and b 0 = (1 /d , . . . , 1/d ) otherwise.These experts are mixed as follows: let {qk, } be a probability distribution over the

set of all pairs (k, ) of positive integers such that for all k, , qk, > 0. If S n (H (k, ) ) isthe capital accumulated by the elementary strategy H (k, ) after n periods when startingwith an initial capital S 0 = 1 , then, after period n , the investors capital becomes

S n (B ) =k,

qk, S n (H (k, ) ) . (5.2)

Gy or, Lugosi and Udina [10] proved that the kernel-based portfolio scheme Bis universally consistent with respect to the class of all ergodic processes such thatE

{| log X ( j )

|} < , for j = 1 , 2, . . . d .Equation (5.1) can be formulated in an equivalent form:

h (k, ) (x n 11 ) = arg maxb d { iJ n }

log b , x i .

Next we introduce the kernel-based Markowitz-type experts H (k, ) = {h(k, ) ()} as

follows:

h (k, ) (xn 11 ) = argmax

b d(1 2)

{ iJ n }

( b , x i 1) { iJ n }

( b , x i 1)2

+

|J n | { iJ n }b , x i

2

. (5.3)

The Markowitz-type kernel-based strategy B is the mixture of the experts {H (k, )

}according to (5.2).

Theorem 5.1 For any stationary and ergodic process {X n } , for which (4.1) holds, for S n, = S n (B ) and for all 0, 12 we get

liminf n

1n

S n,

W +2a 1

1 2

E minm

E 1 + log( X (m )0 ) X (m )0 X

1

2 121 2

E maxm

E (X (m )0 1)2 X 1 minm

E |X (m )0 1| X 1

2

a 3 + 1

3(1 2)E max

mE X (m )0 1

3X 1 a.s.


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Remark 5.2 Under the assumption of Theorem 5.1 with slight modication of the proof one can show same result for all 12 , .

6 SimulationThe theoretical results assume stationarity and ergodicity of the market. No test is known,which could decide whether a market satises these properties or not. Therefore thepractical usefulness of these assumptions should be judged based on the numerical resultsthe investment strategies lead to.

We tested the investment strategies on a standard set of New York Stock Exchangedata used by Cover [8], Singer [18], Hembold, Schapire, Singer, and Warmuth [14], Blumand Kalai [4], Borodin, El-Yaniv, and Gogan [5], and others. The NYSE data set includesdaily prices of 36 assets along a 22-year period (5651 trading days) ending in 1985.

We show the wealth achieved by the strategies by investing in the pairs of NYSEstocks used in Cover [8]: Iroqouis-Kin Ark, Com. Met.-Mei. Corp., Com. Met-Kin Ark and IBM-Coca-Cola. The results of the simulation are shown in the Table 6.1.

All the proposed strategies use an innite array of experts. In practice we take a nitearray of size K L . In all cases select K = 5 and L = 10 . We choose the uniformdistribution {qk, } = 1 / (KL ) over the experts in use, and the radius

r 2k, = 0 .0001 d k ,

(k = 1 , . . . , K and = 1 , . . . , L ).Table 6.1 shows the performance of the nonparametric kernel based Markowitz-type

strategy for different values of parameter . Note that the Table 6.1 contains a rowwith parameter = 0 .5 which is not covered in the Theorem 5.1, however kernel-basedMarkowitz-type strategy can be used in this case too. The k and parameters of the best performing experts given in columns 2 5 are different: (2,10), (3,10), (2,8) and (1,1)respectively. The best performance of pairs of stocks was attained at different values of parameter (underlined in Table 6.1). The average in Table 6.1 means the performanceof the nonparametric kernel based Markowitz-type strategies averaging through . Log-optimal and semi-log-optimal denote the performance of the best performing k and expert of the log-optimal investment and of the semi-log investment for the given pairsof stocks.

Note that the wealth is achieved by the Markowitz-type strategy with parameter value = 0 is not the best overall, although it does not consider the market risk of stocks whichit invests in.

Note that the presence of the stock Kin Ark makes the wealth of these strategiesexplode as it was noted in [10]. This is interesting, since the overall growth of Kin Ark in the reported period is quite modest. The reason is that somehow the variations of the

price relatives of this asset turn out to be well predictable by at least one expert and thatsufces to produce this explosive growth.


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Markowitz S n (H (2,10)

) S n (H(3 ,10) ) S n (H

(2 ,8) ) S n (H

(1 ,1) )

Pairs of stocks. Iro-Kin Com-Mei Com-Kin IBM-Cok

=0.00 2.61e+11 7.13e+03 1.75e+12 1.52e+020.05 2.75e+11 6.73e+03 1.73e+12 1.57e+020.10 2.51e+11 5.74e+03 1.76e+12 1.62e+020.15 2.45e+11 5.44e+03 1.61e+12 1.67e+020.20 2.60e+11 5.87e+03 1.56e+12 1.69e+020.25 2.97e+11 6.12e+03 1.54e+12 1.72e+020.30 3.09e+11 5.66e+03 1.57e+12 1.73e+020.33 3.26e+11 5.47e+03 1.75e+12 1.73e+020.35 3.32e+11 5.46e+03 1.67e+12 1.73e+020.40 3.76e+11 5.45e+03 1.70e+12 1.73e+020.45 3.62e+11 5.12e+03 1.85e+12 1.79e+020.50 3.44e+11 4.82e+03 1.92e+12 1.83e+020.55 3.23e+11 4.07e+03 1.74e+12 1.88e+020.60 2.64e+11 3.28e+03 1.49e+12 1.94e+020.65 2.05e+11 2.62e+03 1.12e+12 2.04e+020.70 1.49e+11 1.97e+03 7.40e+11 2.13e+020.75 7.30e+10 1.53e+03 3.65e+11 2.20e+020.80 1.80e+10 1.19e+03 9.15e+10 2.23e+020.85 1.18e+09 7.02e+02 4.17e+09 2.05e+020.90 8.68e+06 3.30e+02 2.01e+07 1.70e+020.95 1.73e+04 1.38e+02 5.83e+04 7.59e+01

Average 2.22e+11 4040 1.24e+12 177

Log-optimal S n (H (2 ,10) ) S n (H (3 ,10) ) S n (H (2 ,8) ) S n (H (1 ,1) )3.6e+11 4765 1.9e+12 182.4

Semi-log-opt. S n (H (2 ,10) ) S n (H (3 ,10) ) S n (H (2 ,8) ) S n (H (1 ,1) )3.6e+11 4685 1.9e+12 182.6

Table 6.1 Wealth achieved by the strategies by investing in the pairs of NYSE stocks used inCover [8].

7 Proofs

The proof of Theorem 4.1 uses the following two auxiliary results and four other lemmas.The rst is known as Breimans generalized ergodic theorem.

Lemma 7.1 (BREIMAN [6]). Let Z = {Z i } be a stationary and ergodic pro-cess. For each positive integer i , let T i denote the operator that shifts any sequence{. . . , z 1 , z0 , z1 , . . .} by i digits to the left. Let f 1 , f 2 , . . . be a sequence of real-valued functions such that limn f n (Z ) = f (Z ) almost surely for some function f . Assume


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that E {sup n |f n (Z )|} < . Then

limn 1n

n

i =1f i (T

i

Z ) = E {f (Z )} a.s.

The next lemma is a slight modications of the results due to Algoet and Cover [3,Theorems 3 and 4]. The modied statements are in Gy or, Urb an and Vajda [12].

Lemma 7.2 Let Q nN{} be a family of regular probability distributions over the set R d+ of all market vectors such that a U

( j )n 1a for any coordinate of a random market

vector where 0 < a < 1 and U n = ( U (1)n , . . . , U (d )n ) distributed according to Q n .

Let h, g C 0[a, 1a ] , where C 0 denotes the set of continuous functions. In addition, let B (Q n ) be the set of all Markowitz-type portfolios with respect to Q n , that is, the set of all portfolios b that attain max b d {E Q n {h b , U n }+ E 2Q n {g b , U n }}. Consider

an arbitrary sequence b n B

(Q n ). If Q n Q weakly as n

then, for Q -almost all u ,

limn

b n , u b, u

where the right-hand side is constant as b ranges over B (Q ).

For the proof of Theorem 4.1 we need the following lemmas:

Lemma 7.3 For any stationary and ergodic process {X n } , for which (4.1) holds and pC 0[a, 1a ] , we get

limn

1n

n

i =1

maxj

E p X ( j )i Xi 11 = E maxj

E p X ( j )0 X 1 a.s.

Proof. Let us introduce notations

wndef = max

jE p(X ( j )0 ) | X

1 n +1

where n = 1 , 2, . . . and

g(X n )def = p(X (1)n ), . . . , p(X (

d )n ) .

Note thatmaxb d

E b (X 1 n +1 ) , g(X 0) | X 1 n +1 = wn (7.1)

and wn is measurable with respect to X 1 n +1 .


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First we show that {wn } is a sub-martingale, that is, E wn +1 | X 1 n +1 wn . If aportfolio is X 1 n +1 -measurable, then it is also X

1 n -measurable, therefore we obtain

wn = maxb d

E b , g(X 0) |X 1 n +1

= maxb d

E E b , g(X 0) |X 1 n | X 1 n +1

E maxb d

E b , g(X 0) |X 1 n | X 1 n +1

= E wn +1 | X 1 n +1 ,

where in the last equation we applied formula (7.1). Thus wn is a submartingale andE |wn |+ , because of p C 0 a, 1a . Then we can apply convergence theorem of submartingales and we conclude that there exists a random variable w such that

limn

wn = w a.s.

We apply Lemma 7.1 with f i (X )def = wi (X ) parameter, then we get

limn

1n

n

i =1

maxj

E p(X ( j )i )|Xi 11 = E maxj

E p(X ( j )0 )|X 1 a.s.

because of f i (T i X ) = wi (T i X ) = max

jE p(X ( j )0 ) | X

i 11 .

and E {sup i |f i (X ) |} < . The latter one follows from that p() is bounded.

Lemma 7.4 Let Z 1 , . . . , Z n a sequence of random variables then we get the followingupper and lower bound for the logarithmic function of Z n if 0 < 12

U M (Z n , ) + g(Z n , ) E 2{Z n |Z n 11 } +1

3Z 3n(Z n 1)3

1 2 log Z n

U M (Z n , ) + g(Z n , ) E 2{Z n |Z n 11 } + 13 (Z n 1)

3

1 2

where

g(Z n , ) = 2 12

(Z n 1)2 1 + .

Proof. To show the relationship between the log- and the Markowitz-utility function we

use the Taylor expansion of the logarithmic function

U M (Z n , ) (1 2)log Z n

=12

2 (Z n 1)2 + 1 + E 2{Z n |Z n 11 } + R2 , (7.2)


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where R2 = (Z n 1) 33( Z )3 (Z

n [min {Z n , 1}, max {Z n , 1}]) is the Lagrange remainder.

Then using1

3Z 3n (Z n 1)3

R2 13(Z n 1)

3

we obtain the statement of the lemma.

Lemma 7.5 Let X a random market vector satisfying (4.1). Then for any b and b

portfolios

( b , X 1)3

b , X 3 ( b , X 1)3 (a 3 + 1)max

m|X (m ) 1|3 .

Proof. First we show

( b , X 1)3

b , X 3 ( b , X 1)3 (a 3 + 1)max

m|X (m ) 1|3 . (7.3)

If b , X < b , X then

( b , X 1)3

b , X 3 ( b , X 1)3

= ( b , X 1)3

b , X 3 ( b , X 1)3

| b , X 1|3

b , X 3 | b , X 1|3

max | b , X 1|3 , | b , X 1|3 ( b , X 3 + 1) . (7.4)

Let bound the terms in the maximum by Jensens inequality,

| b , X 1|3 =d

m =1b(m ) (X (m ) 1)

3

d

m =1b(m ) X (m ) 1

3 max

mX (m ) 1

3,

(7.5)use b , X 3 a 3 and plug these bounds into (7.4) we obtain (7.3).If b

, X b

, X then

( b , X 1)3

b , X 3 ( b , X 1)3 b , X 3 1 ( b , X 1)3

= b , X 3 1 | b , X 1|3

a 3 1 maxm

X (m ) 13

a 3 + 1 maxm

X (m ) 13

.


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914 Ottucs ak Vajda

With similarly argument we obtain

( b , X 1)3

b , X 3 ( b

, X 1)3

(a 3

+ 1) maxm X (m )

1

3

.

Corollary 7.6 (of Lemma 7.5) Let {X n } be a stationary and ergodic process, sat-isfying (4.1). Then for any b and b portfolios

E( b , X n 1)3

b , X n 3 ( b , X n 1)3 X n 11

(a 3 + 1)maxm

E |X (m )n 1|3 Xn 11 .

Proof. In the proof of Lemma 7.5 instead of equation (7.5) use the following

E | b , X n 1|3 |X n 11 = Ed

m =1b(m ) (X (m )n 1)

3

X n 11

d

m =1b(m ) E X (m )n 1

3X n 11

maxm

E X (m )n 13

X n 11 .

Corollary 7.7 (of Lemma 7.5) Let X be a random market vector satisfying (4.1). Then for any b and b portfolios

( b , X 1)3

b , X 3 E ( b , X 1)3 (a 3 + 1) max

mX (m ) 1

3.

Proof. In the proof of Lemma 7.5 instead of considering cases b , X < b , X andb , X b , X , we split according to b , X < E { b , X } and b , X

E { b , X }. The proof of the two cases goes on the same way as in Lemma 7.5.

Lemma 7.8 Let {X n } be a stationary and ergodic process then

E b(X n 11 ) , X n b , X n |Xn 11 minm

E 1 + log( X (m )n ) X (m )

n | Xn 11 ,

where b(X n 11 ) is the log-optimal portfolio and b d is an arbitrary portfolio.


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Proof. Let us bound from below the rst term of the statement

E b(X n 11 ) , X n | Xn 11 e

E {log b (X n 11 ) , X n | X n 11 } (7.6)

eE {log b (Xn 11 ) , X n | X n

11 } (7.7)

1 +d

m =1b(m ) E log X (m )n | X

n 11 , (7.8)

where (7.6) follows from the Jensen inequality, (7.7) comes from the denition of band(7.8) because of ex 1 + x. Plug this into the left side of the statement we get

E b (X n 11 ) , X n b (Xn 11 ) , X n | X

n 11

d

m =1b(m ) E 1 + log X (m )n X (

m )n | X

n 11

minm E 1 + log X (m

)n X (m

)n | Xn

11

because of E 1 + log X (m )n X (m )n X n 11 0.

We are now ready to prove Theorem 4.1. For convenience in the proof of both theo-rems we use the notations b instead of b , h (k, ) instead of h

(k, ) , B instead of B and

S n instead of S n, .Proof of Theorem 4.1. The parameter is xed. Use Lemma 7.4 with Z n

def = b(X n 11 ) , X n ,where b (X n 11 ) is the Markowitz-type portfolio, then we get

(1 2)E log b (X n 11 ) , X n | Xn 11 E g( b

(X n 11 ) , X n , ) | Xn 11

+ E 2

{b

(Xn 11 ) , X n | X

n 11 }

13

Eb(X n 11 ) , X n 1

3

b(X n 11 ) , X n3 X

n 11

E U M ( b(X n 11 ) , X n , ) | Xn 11

E U M ( b(X n 11 ) , X n , ) | Xn 11

(1 2)E log b (X n 11 ) , X n | Xn 11 E g( b

(X n 11 ) , X n , ) | Xn 11

+ E 2{ b(X n 11 ) , X n | Xn 11 }

13

E ( b (X n 11 ) , X n 1)3 | X n 11 .

After rearranging the above inequalities, we get

(1 2)E log b(X n 11 ) , X n | Xn 11

(1 2)E log b(X n 11 ) , X n | Xn 11

+ E 2{ b(X n 11 ) , X n | X n 11 } E 2{ b(X n 11 ) , X n | X n 11 }+ E g( b(X n 11 ) , X n , ) g( b

(X n 11 ) , X n , ) | Xn 11

+13

E( b(X n 11 ) , X n 1)3

b(X n 11 ) , X n3 ( b

(X n 11 ) , X n 1)3 X n 11 . (7.9)


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916 Ottucs ak Vajda

Taking the arithmetic average on both sides of the inequality over trading periods 1, . . . , n ,then

1n

n

i =1

E log b (X i 11 ) , X i | X i 11

1n

n

i =1

E log b (X i 11 ) , X i | Xi 11

+1n

n

i =1

E 2{ b (X i 11 ) , X i | Xi 11 } E 2{ b(X

i 11 ) , X i | X

i 11 }

1 2

+1n

n

i =1

E g( b (X i 11 ) , X i , ) g( b(Xi 11 ) , X i , ) | X

i 11

1 2

+1

3n

n

i =1

E ( b (X i 11 ) , X i 1)

3

b (X i 11 ) , X i3 ( b(X i 11 ) , X i 1)3 X

i 11

1 2 . (7.10)

We derive simple bounds for the last three additive parts of the above inequality. First,because of < 12

E 2{ b(X i 11 ) , X i | Xi 11 } E

2{ b(X i 11 ) , X i | Xi 11 }

= E { b(X i 11 ) , X i + b(X i 11 ) , X i | X

i 11 }

E { b(X i 11 ) , X i b(X i 11 ) , X i | X

i 11 }

E { b(X i 11 ) , X i + b(X i 11 ) , X i | X

i 11 }

minm

E 1 + log( X (m )i ) X (m )i | X

i 11

2a 1

minm E 1 + log( X (m )i ) X

(m )i | X

i 11 . (7.11)

Second,

Eg( b(X i 11 ) , X i , ) g( b(X

i 11 ) , X i , )

1 2X i 11

2 121 2

maxm

E (X (m )i 1)2 | X i 11

minm

E |X (m )i 1| X 1

2(7.12)

for all value of . And nally, we use Corollary 7.6,

13(1 2)

E( b(X i 11 ) , X i 1)3

b(X i 11 ) , X i3 ( b

(X i 11 ) , X i 1)3 X i 11

a 3 + 1

3(1 2)max

mE |X (m )n 1|3 |X

i 11 . (7.13)


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Consider the following decomposition

1

nlog S

n= Y n + V n ,

where

Y n =1n

n

i =1

log b (X i 11 ) , X i E log b(X i 11 ) , X i |X

i 11

and

V n =1n

n

i =1

E log b(X i 11 ) , X i |Xi 11 .

It can be shown that Y n 0 a.s., since it is an average of bounded martingale differences.So

liminf n V n = liminf n 1n log S n . (7.14)

Similarly, consider the following decomposition

1n

log S n = Y n + V n ,

where

Y n =1n

n

i =1

log b (X i 11 ) , X i E log b(X i 11 ) , X i |X

i 11

and

V n =1n

n

i =1E log b(X i 11 ) , X i |X i 11 .

Again, it can be shown that Y n 0 a.s. Therefore

limn

V n = limn

1n

log S n . (7.15)

Taking the limes inferior of both sides of (7.10) as n goes to innity and applying equal-ities (7.11), (7.12), (7.13), (7.14), (7.15) and Lemma 7.3, we obtain

W liminf n

1n

log S n

W +

1 22a 1 E min

m

E 1 + log( X (m )0

) X (m )0

X 1

2 121 2

E maxm

E (X (m )0 1)2 X 1

a 3 + 1

3(1 2)E max

mE X (m )0 1

3X 1


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918 Ottucs ak Vajda

as desired.

For the proof of Theorem 5.1 we need following two lemmas. The rst is a simplemodication of Theorem 1 in [10].

Lemma 7.9 Under the conditions of Theorem 5.1, for any k, integers and for any xed

limn

1n

n

i =1

U M h (k, ) (X i 11 ) , X i , = E U M bk, (X

1 k ) , X 0 ,

where bk, (X 1 k ) is the Markowitz-type portfolio with respect to the limit distribution

P (k, )X

1 k

.

Proof. Let the integers k, and the vector s = s 1 k Rdk+ be xed. Let P

(k, )j, s denote

the (random) measure concentrated on {X i : 1 j + k i 0, X i 1i k s r k, }dened by

P (k, )j, s (A) =

{ i :1 j + k i 0, X i 1i k s r k, }I A (X i )

|{i : 1 j + k i 0, X i 1i k s r k, }|, A R d+

where I A denotes the indicator function of the set A. If the above set of X i s is empty,then let P (k, )j, s = (1 ,..., 1) be the probability measure concentrated on the vector (1, . . . , 1).Gyor, Lugosi, Udina [10] proved that for all s , with probability one,

P (k, )j, s P

(k, )s =

PX 0 | X 1 k s r k, if

P ( X 1 k s r k, ) > 0,(1 ,..., 1) if P ( X 1 k s r k, ) = 0

(7.16)

weakly as j where P X 0 | X 1 k s r k, denotes the distribution of the vector X 0conditioned on the event X 1 k s r k, .

By denition, b (k, ) (X 11 j , s) is the Markowitz-type portfolio with respect to theprobability measure P (k, )j, s . Let b

k, (s ) denote the Markowitz-type portfolio with respect

to the limit distribution P (k, )s . Then, using Lemma 7.2, we infer from (7.16) that, as jtends to innity, we have the almost sure convergence

limj

b (k, ) (X 11 j , s) , x 0 = bk, (s) , x 0

for P (k, )s -almost all x 0 and hence for P X 0 -almost all x 0 . Since s was arbitrary, weobtain

limj

b (k, ) (X 11 j , X 1 k ) , x 0 = bk, (X 1 k ) , x 0 a.s. (7.17)

Next we apply Lemma 7.1 for the utility function

f i (x )def = U M h (k, ) (x 11 i ) , x 0 , = U M b

(k, ) (x 11 i , x 1 k ) , x 0 ,


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dened on x = ( . . . , x 1 , x 0 , x 1 , . . . ). Because of f i (X ) < and

limi

f i (X ) = U M ( bk, (X

1 k ) , X 0 , ) a.s.

by (7.17). As n , Lemma 7.1 yields

1n

n

i =1

f i (T i X ) =1n

n

i =1

U M h (k, ) (X i 11 ) , X i ,

E U M bk, (X 1 k ) , X 0 ,

as desired.

Lemma 7.10 Under the conditions of Theorem 5.1 there exists a portfolio b c for allc Cfor which

E 2 { b c , X 0 }= c,

where C def = minm E (X (m )

0 )2

, max m E (X(m )0 )

2and furthermore

lim inf n

1n

n

i =1

E 2 h (k, ) (X i 11 ) , X i Xi 11 C.

Proof. The proof has two steps. First we show, that there exists a portfolio for all c Cwhose expected value is c. Let c Cand E 2 { b c , X 0 }= c then

E 2 b c (X 1 k ) , X 0 =

d

m =1b(

m)c E X

(m )0

2def =

d

m =1b(

m)c em

2

.

Denote m = arg min m em and m = arg max m em then let b c portfolio is the follow-ing b(m

)c + b(m

)c = 1 for all other m b(m )c = 0 . So for all c Cthere exists b(m

)c and

b(m )

c that b(m )

c em + b(m )c em

2= c because of the continuity. As the second step

we have to prove that

lim inf n

1n

n

i =1

E 2 h (k, ) (X i 11 ) , X i Xi 11 C.

It is easy to show from Lemma 7.9 and Lemma 7.1 that as n

1n

n

i =1

E h (k, ) (X i 11 ) , X i |X i 11 E b

k, (X

1 k ) , X 0 ,

from which the statement of the lemma follows with argument of contradiction.


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920 Ottucs ak Vajda

Proof of Theorem 5.1. Without loss of generality we may assume S 0 = 1 , so

liminf n

W n (B ) = liminf n

1

nlog S n (B )

= liminf n

1n

logk,

qk, S n (H (k, ) )

liminf n

1n

log supk,

qk, S n (H (k, ) )

= liminf n

1n

supk,

log qk, + log S n (H (k, ) )

= liminf n

supk,

W n (H (k, ) ) +log qk,

n

supk,

liminf n W n (H (k, ) ) + log qk,n

= supk,

liminf n

W n (H (k, ) ), . (7.18)

Because of Lemma 7.4, we can write

W n (H (k, ) ) =1n

n

i =1

log h (k, ) (X i 11 ) , X i

1

1 21n

n

i =1

U M h (k, ) (X i 11 ) , X i ,

1n

n

i =1

E 2 h (k, ) (X i 11 ) , X i | X i 11

+1n

n

i =1

g h (k, ) (X i 11 ) , X i ,

+1

3n

n

i =1

h (k, ) (X i 11 ) , X i 13

h (k, ) (X i 11 ) , X i3 (7.19)

First, we calculate the limes of the rst term, because of Lemma 7.9, we get

limn

1n

n

i =1

U M h (k, ) (X i 11 ) , X i , = E U M bk, (X

1 k ) , X 0 ,

def = k, (7.20)

where bk, (X 1 k ) is a Markowitz-type portfolio with respect to the limit distribution

P (k, )X

1 k

. Let b k, (X 1 k ) denote a log-optimal portfolio with respect to the limit distribu-


21/23


tion P (k, )X

1 k

. Then

k, = E U M bk, (X 1 k ) , X 0 ,

E U M bk, (X 1 k ) , X 0 ,

(1 2)E log bk, (X 1 k ) , X 0 + E2 bk, (X

1 k ) , X 0

E g bk, (X 1 k ) , X 0 , 13

E bk, (X 1 k ) , X 0 1

3(7.21)

where last inequality follows from Lemma 7.4. Let us k,def = E log bk, (X

1 k ) , X 0 .

Combine (7.19), (7.20) and (7.21), then we obtain

liminf n

W n (H (k, ) )

k, + lim inf n

ni =1 E2 b k, (X

1 k ) , X 0 E

2 h (k, ) (X i 11 ) , X i | Xi 11

n(1 2)

+liminf n

ni =1 g h

(k, ) (X i 11 ) , X i , E g bk, (X 1 k ) , X 0 ,

n(1 2)

+liminf n

ni =1

( h ( k, ) (X i 11 ) , X i 1)3

h ( k, ) (X i 11 ) , X i3 E bk, (X

1 k ) , X 0 1

3

3n(1 2).

Now bound the three additive terms separately. First,

E 2 bk, (X 1 k ) , X 0 lim inf n

1n

n

i =1

E 2 h (k, ) (X i 11 ) , X i | Xi 11

= E 2 bk, (X 1 k ) , X 0 E

2 { b , X 0 } (7.22)

= E bk, (X 1 k ) , X 0 + b , X 0 E b k, (X 1 k ) , X 0 b , X 0 E bk, (X 1 k ) , X 0 + b

, X 0

E minm

E 1 + log( X (m )0 ) X (m )0 | X

1 (7.23)

2a 2 E minm

E 1 + log( X (m )0 ) X (m )0 X

1 ,

where (7.22) is true because of Lemma 7.10 ( b is a x portfolio vector). (7.23) followsfrom Lemma 7.8.

For the second and third term we use (7.12) and Corollary 2 of Lemma 7.5 and alsostationarity then we get,

liminf n

W n (H (k, ) ) k, +2a 1

1 2E min

m

E 1 + log( X (m )0

) X (m )0

X 1

2 121 2

E maxm

E (X (m )0 1)2 X 1

a 3 + 1

3(1 2)E max

mE X (m )0 1

3X 1 . (7.24)


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922 Ottucs ak Vajda

Gyor, Lugosi and Udina [10] proved that

supk,

k, = W ,

therefore, by (7.18) and (7.24) the proof of the theorem is nished.

References[1] P. Algoet. Universal schemes for prediction, gambling, and portfolio selection.

Annals of Probability , 20:901941, 1992.

[2] P. Algoet. The strong law of large numbers for sequential decisions under uncer-tainity. IEEE Transactions on Information Theory , 40:609634, 1994.

[3] P. Algoet and T. Cover. Asymptotic optimality asymptotic equipartition propertiesof log-optimum investments. Annals of Probability , 16:876898, 1988.

[4] A. Blum and A. Kalai. Universal portfolios with and without transaction costs. Machine Learning , 35:193205, 1999.

[5] A. Borodin, R. El-Yaniv, and V. Gogan. On the competitive theory and practiceof portfolio selection (extended abstract). In Proc. of the 4th Latin American Sym- posium on Theoretical Informatics (LATIN00) , pages 173196, Punta del Este,Uruguay, 2000.

[6] L. Breiman. The individual ergodic theorem of information theory. Annals of

Mathematical Statistics , 28:809811, 1957. Correction. Annals of MathematicalStatistics , 31:809810, 1960.

[7] L. Breiman. Optimal gambling systems for favorable games. In Proc. of theFourth Berkeley Symposium on Mathematical Statistics and Probability , pages 6578, Berkeley, 1961. University of California Press.

[8] T.M. Cover. Universal portfolios. Mathematical Finance , 1:129, 1991.

[9] R. Grauer. A comparison of growth optimal investment and mean variance invest-ment policies. Journal of Financial and Quantiative Analysis , 16:121, 1981.

[10] L. Gy or, G. Lugosi, and F. Udina. Nonparametric kernel-based sequential invest-

ment strategies. Mathematical Finance , 16:337357, 2006.[11] L. Gy or and D. Schafer. Nonparametric prediction. In J. A. K. Suykens,

G. Horv ath, S. Basu, C. Micchelli, and J. Vandevalle, editors, Advances in Learn-ing Theory: Methods, Models and Applications , pages 339354. IOS Press, NATOScience Series, 2003.


23/23


[12] L. Gy or, A. Urb an, and I. Vajda. Kernel-based semi-log-optimal empirical portfo-lio selection strategies, 2006. Accepted to International Journal of Theoretical andApplied Finance.

[13] N. H. Hakansson and W. T. Ziemba. Capital growth theory. In R. Jarrow, V. Mak-simovic, and W. Ziemba, editors, Handbooks in Operations Research and Manage-ment Science: Finance , volume 9, pages 6186. Elsevier Science, North-Holland,Amsterdam, 1995.

[14] D. P. Helmbold, R. E. Schapire, Y. Singer, and M. K. Warmuth. On-line portfolioselection using multiplicative updates. Mathematical Finance , 8:325344, 1998.

[15] Y. Kroll, H. Levy, and H. Markowitz. Mean variance versus direct utility maxi-mization. Journal of Finance , 39:4775, 1984.

[16] H. Markowitz. Portfolio selection. Journal of Finance , 7(1):7791, 1952.

[17] R. C. Merton. An intertemporal capital asset pricing model. Econometrica ,41(5):867887, 1973.

[18] Y. Singer. Switching portfolios. International Journal of Neural Systems , 8:445455, May 1997.

[19] James Tobin. Liquidity preference as behavior towards risk. The Review of Eco-nomic studies , (67):6586, Feb. 1958.

Gyorgy Ottucs ak Department of Computer Science andInformation Theory

Budapest University of Technologyand EconomicsMagyar Tudosok k orutja 2H1117 Budapest, [email protected]

Istv an VajdaDepartment of Computer Science andInformation Theory

Budapest University of Technologyand EconomicsMagyar Tudosok k orutja 2H1117 Budapest, [email protected]